A method to prove that a modular Galois representation has large image

Abstract

Let $\rho$ be a mod $\ell$ Galois representation attached to a newform $f$. Explicit methods are sometimes able to determine the image of $\rho$, or even the number field cut out by $\rho$, provided that $\ell$ and the level $N$ of $f$ are small enough; however these methods are not amenable to the case where $\ell$ or $N$ are large. The purpose of this short note is to establish a sufficient condition for the image of $\rho$ to be large and which remains easy to test for moderately large $\ell$ and $N$.